Qualitative Treatment of Locus
A 10.1.3
Illustrating Locus with Sketchpad
This activity is to introduce some common locus with the help of Sketchpad.
In this activity, students are required to figure it out by drawing the locus on a paper
before using any IT support.
Activity 1
1.
Ask students to draw a movable chord on a circle with one end (B) fixed on the
circumference while the other end can be moved freely on the circumference.
Study the locus of the midpoint (D) of the chord.
A similar diagram will form like the following.
2.
For less able students, who can draw the locus on the paper but cannot notice
that it is a circle, teachers can introduce the file A10_1_3A.gsp to them.
3.


C can be dragged along the circle.
Students continue to mark more points of D as they reposition C.

All points marked would be shown afterwards to reveal the pattern.
For less able students who cannot get the locus, please refer to the movie
A10_1_3A.mov.
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A 10.1.3
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Qualitative Treatment of Locus
4.
For more able students, ask them to justify if the locus is a circle, or an ellipse.
There are 2 proofs provided in the following.
Fig. 1
i.
Fig. 2
Refer to Fig. 1.
To prove: the locus of the mid-point is a circle
Construction: Show two sets of auxiliary lines to facilitate further investigation
of the locus AD and AB.
(1) Given A is the centre of the circle and A is lying on the locus of D.
(2) For any chord CB.
AD  CB (bisector of chord form centre)
Hence, the locus of D is a circle (converse of angle in semi-circle).
ii. Refer to Fig. 2.
To prove: the locus of the mid-point is a circle
Construction: AC, AB and DE, where E is the mid-point of AB.
(1)
DBE = CBA (common)
(2)
BC BA 2


BD BE 1
(3)
ACB ~ EDB (2 sides proportional and included )
(4)
AC 2

(corr. sides of similar s)
ED 1
As AC is the radius of the circle and is fixed
∴ ED is fixed
∴ D is moved at its fixed distance from E, hence, the locus of D is a circle.
Teachers can show these 2 proofs by illustrating the file A10_1_3A.gsp.
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A 10.1.3
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Qualitative Treatment of Locus
Activity 2
1.
A ladder AB is leaning against the wall and sliding away from it. Study the locus
of the midpoint of the ladder.
A similar figure should be obtained like the following one.
2.
For less able students, who can draw the locus but cannot identify the type of the
path, teachers can introduce the file A10_1_3B.gsp:
For each position of A, roughly locate and mark the mid-point C. After gathering a
number of points, show them all to reveal the pattern, compare with the actual
locus.
3.
Adjust the length of the ladder and start over.
4.
For less able students who cannot draw the locus, teachers can show the movie
to them A10_1_3B.mov.
5.
For more able students, ask them whether the locus is a circular arc or a smooth
path.
The outline of the proof is provided in the following.
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A 10.1.3
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Qualitative Treatment of Locus
D
B
C
O
A
Construction: Show the rectangle with AB as diagonal; another diagonal passing
through the center C; highlights the segment CO.
Reason:
AC constant length. AC = OC by the properties of rectangle. Thus,
OC is constant.
Teachers can show the proof by illustrating the file A10_1_3B.gsp.
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A 10.1.3
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Qualitative Treatment of Locus
Activity 3
1. Enclose a rectangular field against two perpendicular walls with fences BC and CD.
Total length of fences (BC + DC) is fixed. Investigate the locus of the corner C.
A similar figure will be obtained like the following one.
2. For less able students, who can draw the locus but cannot identify the type of the
path, teachers can introduce the file A10_1_3C.gsp:
Drag B to change the rectangle and trace C.
3. For more able students, ask if the locus is a straight line or a smooth path. The
outline of the proof is provided in the following.
Construction: Show triangle AFE (FE is the locus). In order to locate F and E, we
can consider the rectangle being degenerated to a straight line. The vertical point
of this straight line is point F, while the horizontal one is E.
Reason: No matter where C is, ΔDFC and ΔBCE are isosceles triangles. That is
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A 10.1.3
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Qualitative Treatment of Locus
∠BEC = 45°. Thus, the slope of the line being kept constant and the locus passing
through F and E implies the locus is a straight line.
4. Similar to the previous one but this time the area is fixed instead of the lengths.
Repeat Questions 1, 2, 3 in this case. Students may refer to A10_1_3C2.gsp.
© Education and Manpower Bureau 2003
A 10.1.3
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Qualitative Treatment of Locus
Activity 4
1.
A square rolling on the ground. Trace one of its vertexes.
In this part, teachers should ask students to use some real objects, such as a
dice, to roll on the table in order to trace the locus.
A similar figure can be obtained like the following one.
A
A
A
2.
A square rolling on the ground, trace the locus of its centre.
It is suggested to let students use a dice with the face “one” facing them, and
trace the locus of “one”.
A similar figure may be obtained like the following one.
© Education and Manpower Bureau 2003
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